Partition Of An Integer Calculator

Work out partition of an integer instantly with clear inputs, formula shown and shareable results.

About Partition Of An Integer Calculator

A partition writes n as an unordered sum of positive integers. The count p(n) is computed by dynamic programming over allowed part sizes, and restricting the largest part or requiring distinct parts gives smaller counts.

Formula

Partition recurrence

build p by iterating allowed parts and accumulating p[v] += p[v − part]

Worked example

With the default values below, Partition Of An Integer Calculator returns Number of partitions p(n) of 42. Change any input above and every figure here is recalculated in your browser.

Inputs used

Integer to partitionaccepts 0–200: 10
Restrict parts to at mostaccepts 1–200: 4

Results

Number of partitions p(n): 42
Partitions with parts at most your limit: 23
Partitions into distinct parts: 10
Integer partitioned: 10

Step by step

1Values used
2Partition recurrence
3Number of partitions p(n)= 42
4Partitions with parts at most your limit= 23
5Partitions into distinct parts= 10
6Integer partitioned= 10

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How many partitions does 10 have?

42, of which 10 use parts no larger than 4 and 10 use distinct parts.

Is there a closed formula?

No simple one. Hardy and Ramanujan gave an asymptotic p(n) ~ exp(π√(2n/3))/(4n√3), later made exact by Rademacher's series.